On the minimum path problem in Knodel graphs

The Knodel graphs W-d,W-n are regular graphs, of even order n and degree d <= [log n]. They have been introduced by W. Knodel as gossip graphs for d = [log n]. A logarithmic algorithm for the minimum path problem in Knodel graphs is an open problem despite the fact that they are bipartite and highly symmetric. In this paper, we describe a logarithmic time two-approximation algorithm for the shortest path in the Knodel graph on 2(d) vertices with degree d. We also prove that for a subset of the set of vertices, the algorithm gives a minimum length path. (C) 2007 Wiley Periodicals, Inc.